2.1. What are bonds?

and solutions occurring in BPI, but also selected mathematical concepts and tools which proved to be instrumental in developing BPI. I do believe that such information has a good chance to be useful in creation of immunity against particular diseases. Bond investors are called immunizers if, possessing C dollars today, they must achieve an investment goal of L dollars q years from now (a human organism or a particular human organ must achieve a certain level of health q years from now); here L is the future value of C at time q under the current interest rates. This investment goal must be accomplished by means of an appropriately selected bond portfolio, even despite unfavorable sudden change (shift) in interest rates (appearance of a disease), having in mind that the present and future prices of all bonds

Although, as it will be demonstrated in Sections 3.1, 3.2, and 3.3, immunization against all shifts is never possible, there are many results giving sufficient, or necessary and sufficient, conditions for immunization against a certain classes of shifts (certain diseases). It is worth to know that in the financial immunization, there is no such thing as acquired immunity (immunity that develops in a human after exposure to a suitable agent) and active immunity (acquired through production

Such types of immunization might theoretically take place on a bond market only if a bond holder had the right to change the coupon payments, which is completely out of the question. In other words, the immunization in financial reality has features of passive immunity, being in fact a short-acting immunity. On the other hand, however, a BP manager can achieve the state of a BP being all the time immune against a specific class of shifts, provided the manager regularly (every week or so) performs (if necessary) subsequent adjustments of his/her BP

Theorem 1 (Section 2.4.2) as well as Theorems 3 and 4 allows one to look at immunization from a different perspective. They enable one to identify all shifts a(t) (diseases) of the term structure s(t) of interest rates against which BP is already (fully) immunized, that is, protected against loss of its value at time q. Finally, having identified immunized (immune) bond portfolios, the natural question arises how to find among them the best ones. This topic is dealt with in Section 4.

Below, I shall present (i) what are bonds and bond portfolios; (ii) what is meant by standard (and general) immunization problem; (iii) historical development of immunization theory; (iv) overview of some recent results; (v) the concept of a Hilbert space, and a base in a linear space; (vi) application of orthogonal polynomials to description of the class IMMU of all shifts (diseases) against which a given bond portfolio is immunized; (vii) triangular functions as a base for the linear space IMMU; and (viii) the crucial role of the notions of duration and

Below, we will introduce the concept of bonds, formulate the standard and general immunization problem, and outline the development of immunization theory in finance, from the

of antibodies within the organism in response to the presence of antigens).

according to their expertise in the area of immunization theory.

convexity in choosing the "best" immunized (immune) bond portfolios.

depend solely on interest rates.

96 Immunization - Vaccine Adjuvant Delivery System and Strategies

2. Immunization in finance

beginning to the latest achievements.

Each bond with a face value (par value) of F dollars is a financial instrument which generates to its buyer (holder) specified payments every 6 months (or every quarter, every year) in the form of coupons, plus par value paid only at the termination (maturity) of the bond. The face value represents the amount borrowed by the seller (issuer) of a bond from the bond buyer. The coupons represent a predetermined percentage, say 3%, of face value F; if F = \$10,000, then all coupons paid per year sum up to \$300. Each bond has its own life span (maturity) of n years (3, 5, 10, 20 years, etc.)

A bond portfolio (BP), by its definition, is a collection of different bonds with various maturities. Thus, each BP generates a more complicated cash flow pattern than a single bond does. A cash flow generated by a BP consists of various size payments ci, 1 ≤ i ≤ m, (coupons and par values generated by all kinds of bonds forming that portfolio) at certain dates t1, t2, t3, …, tm from an interval ½ � t0; T , where t<sup>0</sup> is the date when BP was purchased, while T stands for the highest maturity of all bonds tradable on a given debt market D.

The present and future value of each bond, and consequently each bond portfolio, depends solely on current interest rates s tð Þ, which in the simplest case are identical for all maturities t, that is, s tðÞ� s, t∈½ � t0; T . By the term structure of interest rates, one understands a schedule of spot interest rates s tð Þ which are estimated from the yields (returns) of all coupon-bearing bonds. It is well known that interest rates are shaped under various random market forces.

### 2.2. Standard and general immunization problem

The standard immunization problem relies on a construction of such a bond portfolio with the present value of C dollars that the single liability to pay L dollars q years from now (L is the future value of C) by means of the cash flow generated by BP will be secured regardless of how adverse changes in interest rates will occur in a future. This nontrivial problem is automatically solved by each zero-coupon bond maturing at time q with par value of L dollars. Thus, using medical terminology, one may say that such a zero-coupon bond possesses an innate (natural) immunity. Unfortunately, in practice, such zero-coupon bonds rarely exist.

Besides, an investor may already possess bonds and would like to buy additional ones so that the created, in this way, portfolio BP with the present value of C dollars would secure the payment of L dollars q years from now. Having built such a portfolio, the investor would immunize (hedge) their own investment against a loss of its value at time q. We assume that the new term structure will always be of the form s\*(t) = s(t) + a(t), where a(t) belongs to a certain class of shifts (diseases).

On the other hand, the general immunization problem relies on a construction of such a bond portfolio BP with the present value of C dollars that multiply liabilities to pay Li dollars at specified instances of time will be secured by means of the cash flow generated by BP regardless of adverse changes/shifts a(t) of interest rates in a future.

#### 2.3. Beginnings of immunization

Immunization as a concept dates back as far as to articles [1, 2]. However, not until work [3] of Fisher and Weil was the impact of interest shifts on the design of immunization strategies

rigorously studied. In a vast majority of publications, immunization was based on a specific stochastic process governing interest rate shifts a(t). In [2], Redington discussed immunization in the context of an actuarial company which had projected liability outflows L(t) at some finite number M of instances (dates) tk and anticipated inflows A tð Þ<sup>i</sup> at N (typically) different dates ti. It was assumed that interest rates were flat, that is, s tðÞ� s, and shocks a(t) of interest rates s(t) meant their parallel movements, that is, a tðÞ� λ.

In such a situation, the company's task was to choose inflows A(t) in such a manner that the outflows L(t) would be discharged if the interest rates s tðÞ� s moved to their new constant

level <sup>s</sup><sup>∗</sup>ðÞ� <sup>t</sup> <sup>s</sup> <sup>þ</sup> <sup>λ</sup>. To recall Redington's main result, let us note that <sup>V</sup> <sup>¼</sup> <sup>t</sup> P¼N t¼1 A tð Þ ð Þ <sup>1</sup>þ<sup>s</sup> <sup>t</sup> represents the present value of inflows A(t) occurring at instances ti; a similar formula holds for liabilities L(t). Redington introduced the notion of a "mean term" having in mind the weighted average of the dates when the flows are to be received (in case of assets) or have to be discharged (in case of liabilities). This "mean term" was nothing different than the concept of duration introduced by Macaulay in [1]. These two authors understood duration as:

$$D = \sum\_{t=1}^{t=N} tw\_t; w\_t = \frac{A(t)}{V(1+s)^t}; \sum\_{t=1}^{t=N} w\_t = 1\tag{1}$$

while the duration DA was given by

are equal to 0 (zero).

of additive stochastic process h<sup>∗</sup>

multiplicative shift

and the additive one

were derived.

DA ¼

2.4.1. Further developments of immunization theory

ð<sup>N</sup> 0

duration of the single liability to be discharged at time <sup>q</sup>: DA <sup>¼</sup> <sup>Ð</sup> <sup>N</sup>

wttdt ¼

made so far that interest rates were subject to random shifts of the form h<sup>∗</sup>

the resulting duration differ depending on the underlying stochastic process.

ð Þ¼ 0; t 1 þ

ð Þ¼ 0; t hð Þþ 0; t

interest rate movements (diseases) as long as his/her predictions are accurate.

example, when a multiplicative stochastic process λ is used, that is, h<sup>∗</sup>

h∗

h∗

2.4.2. Latest developments of immunization theory

admissible shifts was analyzed in [10].

ð<sup>N</sup> 0

It was stated in [3] that immunization was secured if the duration of the assets DA equaled the

of any single inflow or outflow at time q equals q since all weights, except for the one at date q,

In subsequent 20–30 years of development of immunization theory, the strong assumption

was being dropped. Many authors began to study shifts governed by some specific stochastic processes. For example, it was proved in [4] that some alternative stochastic processes permitted immunization, and others did not. When immunization can occur, formulas for calculating

Few years later, it was demonstrated in [5] that these differences may be significant. For

the immunizing duration shown as Eq. (13) in [6] on p. 29. On the other hand, both the

were studied in [7], where suitable implicit formulas for the respective immunizing durations

Another approach, called contingent immunization, was developed in [8]. It consists of building a bond portfolio with a duration shorter or longer than the investor's planning horizon, taking into account "personal" expectations of a bond manger with regard interest rates. The idea standing behind such approach is to take advantage of the manager's ability to forecast

The contingent immunization was implemented in many situations for various term structures of interest rates; see [9]. Other than mentioned in Eqs. (4) and (5), stochastic process governing

λ ln 1ð Þ þ αt αt

λ ln 1ð Þ þ αt

ð Þ¼ 0; t h 0ð Þþ ; t λ, one would obtain the implicit formula for

� �hð Þ <sup>0</sup>; <sup>t</sup> ; (4)

<sup>α</sup><sup>t</sup> , (5)

A tð Þ exp ½ � �hð Þ 0; t VA

tdt: (3)

Practical Finance Strategies in Immunization http://dx.doi.org/10.5772/intechopen.78719 99

<sup>0</sup> wttdt ¼ q. Clearly, duration

ð Þ¼ 0; t h 0ð Þþ ; t λ

ð Þ¼ 0; t λhð Þ 0; t instead

where wt tells us what portion (weight) of the entire cash flow is represented by A(t) in terms of today's money. It was proved in [2] that any parallel movement (shift) of the flat term structure s tðÞ� s of interest rates would affect the value of the assets in the same way as it would affect the value of liabilities if duration DA of assets A(t) were equal to duration DL of liabilities L(t), and additionally, the so-called convexity of the assets would exceed that of the liabilities.

#### 2.4. Assumptions concerning term structure of interest rates and admissible shifts: historical development

Twenty years later, Fisher and Weil [3] restricted themselves to a single liability at a specified date q, but significantly weakened the adopted so far assumption that the term structure was flat, that is, s tðÞ� s. Denoting current interest rates s(t) as h(0,t), they allowed h(0,t) to be a function of arbitrary shape with h(0,t), 0 ≤ t ≤ N, meaning annualized returns on zero-coupon default-free bonds tradable on a debt market D. However, they upheld the strong assumption concerning the admissible shifts a(t) by supposing that h(0,t) was subject only to a random additive shift of the form h<sup>∗</sup> ð Þ¼ 0; t h 0ð Þþ ; t λ for 0 ≤ t ≤ N appearing instantly after the acquisition of a bond portfolio.

They applied (popular already at that time) continuous compounding of cash flows A(t) and L(t), which in their approach represented instantaneous rate of payments per one unit of time rather than payments themselves, so that the present value of the assets VA could be expressed by means of an integral

$$V\_A = \int\_0^N A(t) \exp\left[-h(0, t)\right] t dt \tag{2}$$

while the duration DA was given by

rigorously studied. In a vast majority of publications, immunization was based on a specific stochastic process governing interest rate shifts a(t). In [2], Redington discussed immunization in the context of an actuarial company which had projected liability outflows L(t) at some finite number M of instances (dates) tk and anticipated inflows A tð Þ<sup>i</sup> at N (typically) different dates ti. It was assumed that interest rates were flat, that is, s tðÞ� s, and shocks a(t) of interest rates s(t)

In such a situation, the company's task was to choose inflows A(t) in such a manner that the outflows L(t) would be discharged if the interest rates s tðÞ� s moved to their new constant

the present value of inflows A(t) occurring at instances ti; a similar formula holds for liabilities L(t). Redington introduced the notion of a "mean term" having in mind the weighted average of the dates when the flows are to be received (in case of assets) or have to be discharged (in case of liabilities). This "mean term" was nothing different than the concept of duration intro-

twt; wt <sup>¼</sup> A tð Þ

where wt tells us what portion (weight) of the entire cash flow is represented by A(t) in terms of today's money. It was proved in [2] that any parallel movement (shift) of the flat term structure s tðÞ� s of interest rates would affect the value of the assets in the same way as it would affect the value of liabilities if duration DA of assets A(t) were equal to duration DL of liabilities L(t), and additionally, the so-called convexity of the assets would exceed that of the liabilities.

2.4. Assumptions concerning term structure of interest rates and admissible shifts: historical

Twenty years later, Fisher and Weil [3] restricted themselves to a single liability at a specified date q, but significantly weakened the adopted so far assumption that the term structure was flat, that is, s tðÞ� s. Denoting current interest rates s(t) as h(0,t), they allowed h(0,t) to be a function of arbitrary shape with h(0,t), 0 ≤ t ≤ N, meaning annualized returns on zero-coupon default-free bonds tradable on a debt market D. However, they upheld the strong assumption concerning the admissible shifts a(t) by supposing that h(0,t) was subject only to a random

They applied (popular already at that time) continuous compounding of cash flows A(t) and L(t), which in their approach represented instantaneous rate of payments per one unit of time rather than payments themselves, so that the present value of the assets VA could be expressed

VA ¼

ð<sup>N</sup> 0

<sup>V</sup>ð Þ <sup>1</sup> <sup>þ</sup> <sup>s</sup> <sup>t</sup> ;

Xt¼N t¼1

ð Þ¼ 0; t h 0ð Þþ ; t λ for 0 ≤ t ≤ N appearing instantly after the acquisi-

A tð Þ exp ½ � �hð Þ 0; t tdt (2)

P¼N t¼1

wt ¼ 1 (1)

A tð Þ

ð Þ <sup>1</sup>þ<sup>s</sup> <sup>t</sup> represents

level <sup>s</sup><sup>∗</sup>ðÞ� <sup>t</sup> <sup>s</sup> <sup>þ</sup> <sup>λ</sup>. To recall Redington's main result, let us note that <sup>V</sup> <sup>¼</sup> <sup>t</sup>

duced by Macaulay in [1]. These two authors understood duration as:

<sup>D</sup> <sup>¼</sup> <sup>X</sup><sup>t</sup>¼<sup>N</sup> t¼1

meant their parallel movements, that is, a tðÞ� λ.

98 Immunization - Vaccine Adjuvant Delivery System and Strategies

development

additive shift of the form h<sup>∗</sup>

tion of a bond portfolio.

by means of an integral

$$D\_A = \int\_0^N w\_l t dt = \int\_0^N \frac{A(t) \exp\left[-h(0, t)\right]}{V\_A} t dt. \tag{3}$$

It was stated in [3] that immunization was secured if the duration of the assets DA equaled the duration of the single liability to be discharged at time <sup>q</sup>: DA <sup>¼</sup> <sup>Ð</sup> <sup>N</sup> <sup>0</sup> wttdt ¼ q. Clearly, duration of any single inflow or outflow at time q equals q since all weights, except for the one at date q, are equal to 0 (zero).

#### 2.4.1. Further developments of immunization theory

In subsequent 20–30 years of development of immunization theory, the strong assumption made so far that interest rates were subject to random shifts of the form h<sup>∗</sup> ð Þ¼ 0; t h 0ð Þþ ; t λ was being dropped. Many authors began to study shifts governed by some specific stochastic processes. For example, it was proved in [4] that some alternative stochastic processes permitted immunization, and others did not. When immunization can occur, formulas for calculating the resulting duration differ depending on the underlying stochastic process.

Few years later, it was demonstrated in [5] that these differences may be significant. For example, when a multiplicative stochastic process λ is used, that is, h<sup>∗</sup> ð Þ¼ 0; t λhð Þ 0; t instead of additive stochastic process h<sup>∗</sup> ð Þ¼ 0; t h 0ð Þþ ; t λ, one would obtain the implicit formula for the immunizing duration shown as Eq. (13) in [6] on p. 29. On the other hand, both the multiplicative shift

$$h^\*(0, t) = \left[1 + \frac{\lambda \ln\left(1 + \alpha t\right)}{\alpha t}\right] h(0, t);\tag{4}$$

and the additive one

$$h^\*(0, t) = h(0, t) + \frac{\lambda \ln\left(1 + \alpha t\right)}{\alpha t},\tag{5}$$

were studied in [7], where suitable implicit formulas for the respective immunizing durations were derived.

Another approach, called contingent immunization, was developed in [8]. It consists of building a bond portfolio with a duration shorter or longer than the investor's planning horizon, taking into account "personal" expectations of a bond manger with regard interest rates. The idea standing behind such approach is to take advantage of the manager's ability to forecast interest rate movements (diseases) as long as his/her predictions are accurate.

#### 2.4.2. Latest developments of immunization theory

The contingent immunization was implemented in many situations for various term structures of interest rates; see [9]. Other than mentioned in Eqs. (4) and (5), stochastic process governing admissible shifts was analyzed in [10].

Striving to offer a more general approach, the authors of article [11] did not confine ourselves to a specific process governing shifts, but allowed them to belong to a certain class of functions, such as, for example, polynomials of degree less than some specified number n (pp. 858–861). In this way, they did not expose themselves to any model misspecification risk, similarly as Zheng in [12].

The larger class of shifts (diseases) against which the immunization will work, the better. Having this in mind, the interval ½ � t0; T was divided in [6] into n equal nonoverlapping subintervals Ik, 1 ≤ k ≤ n, and set akðÞ¼ t 1 when t∈ Ik and akðÞ¼ t 0 otherwise (p. 34). The admissible shifts were assumed to be piecewise constant functions of the form k P¼n k¼1 λkakð Þt . The authors made a general assumption stating that a BP generates inflows

$$A(t) = A\_0(t) + \sum\_{k=1}^{k=m} c\_k \delta(t - t\_k),\tag{6}$$

say IMMU, against which a bond portfolio BP is already immunized, that is, protected against loss of its value at time q. In this context, Theorem 1 offers a sufficient and necessary condition

Practical Finance Strategies in Immunization http://dx.doi.org/10.5772/intechopen.78719 101

As of today, no one was successful in building up a bond portfolio BP immunized (immune) against all shifts of interest rates (diseases). What is more, in Section 3, we demonstrate that the set IMMU is always a proper subset of all admissible shifts, being in fact a linear subspace of all

In a recent paper [13], the authors found a strong evidence that momentum across various asset classes is caused by macroeconomic variables. By properly modifying their portfolio, in response to changes in macroeconomic environment, their strategy performed particularly well in times of economic distress. The obtained results allowed them to establish a link

Aiming at securing higher effectiveness of their investment in fixed income bonds, the authors of [14] successfully used simulations of the portfolio surplus, measuring the inherent risk by means of the value-at-risk methodology. In another very recent publication [15], the authors studied immunization assuming that shifts were parallel or symmetric. A quite different approach to immunization was proposed in [16]. The authors concentrated on hedging risk inherent in bond portfolio. They divided the entire problem into two parts, by formulating a two-step optimization problem. They focused first on immunization risk, and next maximized

In this section, it is proved that the set of all continuous shocks a(t) against which a bond portfolio BP is immunized is an m-dimensional linear subspace in the (m + 1)-dimensional linear space of all continuous shifts a(t), with m standing for the number of instances when BP promises to pay cash (coupons or par values generated by bonds forming BP). The main mathematical concept used below is the notion of a Hilbert space and the concept of a base in a Hilbert space.

From now on, we assume that A0ðÞ� t 0 in Formula 6, so that inflows given by

A tðÞ¼ <sup>X</sup> k¼m

k¼1

generate only payments ck at specified instances t1, t2, t3, …, tm. In such a situation, the present

ckδð Þ t � tk , (9)

2.5. One cannot immunize against all possible shifts of interest rates development

for a shift a(t) to belong to set IMMU.

3. Overview of some recent results

3.1. When polynomials are admissible shifts

value of assets A(t) is no longer given by Eq. (2), but by

between momentum and sophisticated predictive regressions.

shifts.

the portfolio wealth.

with A0ð Þt representing an instantaneous rate of cash payout, while sk standing for single payment at instances t1, t2, t3, …, tm: The expression δð Þ t � tk , with δð Þt standing for a Dirac delta function, was employed in order to make integration possible. The following result (Theorem 3, pp. 34–35 in [6]) was then proved.

Theorem 1. If q denotes the date when the single liability of L dollars has to be discharged by means of the cumulative value of assets A(t), then the immunization is secured against all adverse piecewise constant shifts k P¼n k¼1 λkakð Þt of interest rates h(0,t) if and only if

$$a\_k(q)q = \int\_0^T \frac{A(t) \exp\left[-h(0,t)\right]t}{V\_A} a\_k(t)t dt, \quad 1 \le k \le n,\tag{7}$$

where VA stands for the present value of the portfolio represented by assets A(t).

Remark 1. When n = 1, then Theorem 1 gives a sufficient and necessary condition

$$a(q)q = \int\_0^T \frac{A(t) \exp\left[-h(0,t)\right]t}{V\_A} a(t)t dt\tag{8}$$

for immunization when the term structure h(0,t) is subject to shifts h tð Þþ ; 0 λa tð Þ. In case of parallel shifts, (8) reduces to Fisher-Weil condition <sup>q</sup> <sup>¼</sup> <sup>Ð</sup> <sup>T</sup> 0 A tð Þ exp ½ � �hð Þ 0;t t VA tdt, which is well remembered as the following statement: Immunization is secured if the duration of the assets DA equals the duration of the single liability.

Remark 2. Theorem 1 can also be looked at from a different perspective. Namely, one may be interested in identification of such a set of shifts a(t) of the term structure s(t) of interest rates, say IMMU, against which a bond portfolio BP is already immunized, that is, protected against loss of its value at time q. In this context, Theorem 1 offers a sufficient and necessary condition for a shift a(t) to belong to set IMMU.

#### 2.5. One cannot immunize against all possible shifts of interest rates development

As of today, no one was successful in building up a bond portfolio BP immunized (immune) against all shifts of interest rates (diseases). What is more, in Section 3, we demonstrate that the set IMMU is always a proper subset of all admissible shifts, being in fact a linear subspace of all shifts.
